Theorem isdivrngo 37979

Description: The predicate "is a division ring". (Contributed by FL, 6-Sep-2009.) (New usage is discouraged.)

Assertion

Ref	Expression
isdivrngo	⊢ (𝐻 ∈ 𝐴 → (⟨𝐺, 𝐻⟩ ∈ DivRingOps ↔ (⟨𝐺, 𝐻⟩ ∈ RingOps ∧ (𝐻 ↾ ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)}))) ∈ GrpOp)))

Proof of Theorem isdivrngo

Dummy variables 𝑔 ℎ 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-br 5125	. . . . 5 ⊢ (𝐺DivRingOps𝐻 ↔ ⟨𝐺, 𝐻⟩ ∈ DivRingOps)
2		df-drngo 37978	. . . . . . 7 ⊢ DivRingOps = {⟨𝑥, 𝑦⟩ ∣ (⟨𝑥, 𝑦⟩ ∈ RingOps ∧ (𝑦 ↾ ((ran 𝑥 ∖ {(GId‘𝑥)}) × (ran 𝑥 ∖ {(GId‘𝑥)}))) ∈ GrpOp)}
3	2	relopabiv 5804	. . . . . 6 ⊢ Rel DivRingOps
4	3	brrelex1i 5715	. . . . 5 ⊢ (𝐺DivRingOps𝐻 → 𝐺 ∈ V)
5	1, 4	sylbir 235	. . . 4 ⊢ (⟨𝐺, 𝐻⟩ ∈ DivRingOps → 𝐺 ∈ V)
6	5	anim1i 615	. . 3 ⊢ ((⟨𝐺, 𝐻⟩ ∈ DivRingOps ∧ 𝐻 ∈ 𝐴) → (𝐺 ∈ V ∧ 𝐻 ∈ 𝐴))
7	6	ancoms 458	. 2 ⊢ ((𝐻 ∈ 𝐴 ∧ ⟨𝐺, 𝐻⟩ ∈ DivRingOps) → (𝐺 ∈ V ∧ 𝐻 ∈ 𝐴))
8		rngoablo2 37938	. . . . 5 ⊢ (⟨𝐺, 𝐻⟩ ∈ RingOps → 𝐺 ∈ AbelOp)
9		elex 3485	. . . . 5 ⊢ (𝐺 ∈ AbelOp → 𝐺 ∈ V)
10	8, 9	syl 17	. . . 4 ⊢ (⟨𝐺, 𝐻⟩ ∈ RingOps → 𝐺 ∈ V)
11	10	ad2antrl 728	. . 3 ⊢ ((𝐻 ∈ 𝐴 ∧ (⟨𝐺, 𝐻⟩ ∈ RingOps ∧ (𝐻 ↾ ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)}))) ∈ GrpOp)) → 𝐺 ∈ V)
12		simpl 482	. . 3 ⊢ ((𝐻 ∈ 𝐴 ∧ (⟨𝐺, 𝐻⟩ ∈ RingOps ∧ (𝐻 ↾ ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)}))) ∈ GrpOp)) → 𝐻 ∈ 𝐴)
13	11, 12	jca 511	. 2 ⊢ ((𝐻 ∈ 𝐴 ∧ (⟨𝐺, 𝐻⟩ ∈ RingOps ∧ (𝐻 ↾ ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)}))) ∈ GrpOp)) → (𝐺 ∈ V ∧ 𝐻 ∈ 𝐴))
14		df-drngo 37978	. . . 4 ⊢ DivRingOps = {⟨𝑔, ℎ⟩ ∣ (⟨𝑔, ℎ⟩ ∈ RingOps ∧ (ℎ ↾ ((ran 𝑔 ∖ {(GId‘𝑔)}) × (ran 𝑔 ∖ {(GId‘𝑔)}))) ∈ GrpOp)}
15	14	eleq2i 2827	. . 3 ⊢ (⟨𝐺, 𝐻⟩ ∈ DivRingOps ↔ ⟨𝐺, 𝐻⟩ ∈ {⟨𝑔, ℎ⟩ ∣ (⟨𝑔, ℎ⟩ ∈ RingOps ∧ (ℎ ↾ ((ran 𝑔 ∖ {(GId‘𝑔)}) × (ran 𝑔 ∖ {(GId‘𝑔)}))) ∈ GrpOp)})
16		opeq1 4854	. . . . . 6 ⊢ (𝑔 = 𝐺 → ⟨𝑔, ℎ⟩ = ⟨𝐺, ℎ⟩)
17	16	eleq1d 2820	. . . . 5 ⊢ (𝑔 = 𝐺 → (⟨𝑔, ℎ⟩ ∈ RingOps ↔ ⟨𝐺, ℎ⟩ ∈ RingOps))
18		rneq 5921	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → ran 𝑔 = ran 𝐺)
19		fveq2 6881	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (GId‘𝑔) = (GId‘𝐺))
20	19	sneqd 4618	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → {(GId‘𝑔)} = {(GId‘𝐺)})
21	18, 20	difeq12d 4107	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (ran 𝑔 ∖ {(GId‘𝑔)}) = (ran 𝐺 ∖ {(GId‘𝐺)}))
22	21	sqxpeqd 5691	. . . . . . 7 ⊢ (𝑔 = 𝐺 → ((ran 𝑔 ∖ {(GId‘𝑔)}) × (ran 𝑔 ∖ {(GId‘𝑔)})) = ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)})))
23	22	reseq2d 5971	. . . . . 6 ⊢ (𝑔 = 𝐺 → (ℎ ↾ ((ran 𝑔 ∖ {(GId‘𝑔)}) × (ran 𝑔 ∖ {(GId‘𝑔)}))) = (ℎ ↾ ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)}))))
24	23	eleq1d 2820	. . . . 5 ⊢ (𝑔 = 𝐺 → ((ℎ ↾ ((ran 𝑔 ∖ {(GId‘𝑔)}) × (ran 𝑔 ∖ {(GId‘𝑔)}))) ∈ GrpOp ↔ (ℎ ↾ ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)}))) ∈ GrpOp))
25	17, 24	anbi12d 632	. . . 4 ⊢ (𝑔 = 𝐺 → ((⟨𝑔, ℎ⟩ ∈ RingOps ∧ (ℎ ↾ ((ran 𝑔 ∖ {(GId‘𝑔)}) × (ran 𝑔 ∖ {(GId‘𝑔)}))) ∈ GrpOp) ↔ (⟨𝐺, ℎ⟩ ∈ RingOps ∧ (ℎ ↾ ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)}))) ∈ GrpOp)))
26		opeq2 4855	. . . . . 6 ⊢ (ℎ = 𝐻 → ⟨𝐺, ℎ⟩ = ⟨𝐺, 𝐻⟩)
27	26	eleq1d 2820	. . . . 5 ⊢ (ℎ = 𝐻 → (⟨𝐺, ℎ⟩ ∈ RingOps ↔ ⟨𝐺, 𝐻⟩ ∈ RingOps))
28		reseq1 5965	. . . . . 6 ⊢ (ℎ = 𝐻 → (ℎ ↾ ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)}))) = (𝐻 ↾ ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)}))))
29	28	eleq1d 2820	. . . . 5 ⊢ (ℎ = 𝐻 → ((ℎ ↾ ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)}))) ∈ GrpOp ↔ (𝐻 ↾ ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)}))) ∈ GrpOp))
30	27, 29	anbi12d 632	. . . 4 ⊢ (ℎ = 𝐻 → ((⟨𝐺, ℎ⟩ ∈ RingOps ∧ (ℎ ↾ ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)}))) ∈ GrpOp) ↔ (⟨𝐺, 𝐻⟩ ∈ RingOps ∧ (𝐻 ↾ ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)}))) ∈ GrpOp)))
31	25, 30	opelopabg 5518	. . 3 ⊢ ((𝐺 ∈ V ∧ 𝐻 ∈ 𝐴) → (⟨𝐺, 𝐻⟩ ∈ {⟨𝑔, ℎ⟩ ∣ (⟨𝑔, ℎ⟩ ∈ RingOps ∧ (ℎ ↾ ((ran 𝑔 ∖ {(GId‘𝑔)}) × (ran 𝑔 ∖ {(GId‘𝑔)}))) ∈ GrpOp)} ↔ (⟨𝐺, 𝐻⟩ ∈ RingOps ∧ (𝐻 ↾ ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)}))) ∈ GrpOp)))
32	15, 31	bitrid 283	. 2 ⊢ ((𝐺 ∈ V ∧ 𝐻 ∈ 𝐴) → (⟨𝐺, 𝐻⟩ ∈ DivRingOps ↔ (⟨𝐺, 𝐻⟩ ∈ RingOps ∧ (𝐻 ↾ ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)}))) ∈ GrpOp)))
33	7, 13, 32	pm5.21nd 801	1 ⊢ (𝐻 ∈ 𝐴 → (⟨𝐺, 𝐻⟩ ∈ DivRingOps ↔ (⟨𝐺, 𝐻⟩ ∈ RingOps ∧ (𝐻 ↾ ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)}))) ∈ GrpOp)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3464   ∖ cdif 3928  {csn 4606  ⟨cop 4612   class class class wbr 5124  {copab 5186   × cxp 5657  ran crn 5660   ↾ cres 5661  ‘cfv 6536  GrpOpcgr 30475  GIdcgi 30476  AbelOpcablo 30530  RingOpscrngo 37923  DivRingOpscdrng 37977

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-ov 7413  df-1st 7993  df-2nd 7994  df-rngo 37924  df-drngo 37978

This theorem is referenced by:  zrdivrng  37982  isdrngo1  37985